ON ELLIPTIC AND HYPERELLIPLIC FUNCTIONS. 



359 



ej)- B/-fx-=l"6o—l"Qx, 

 e o" d.fi-ffAr = l"dx---l"d.^O, 

 6 0- ey- hx-=l"ox—l"d..o, 



e.ye,o--±=rdo-rd,x, 



0- 8,0^ — , = i"e,o—i"d,x, y 

 60- ey- ,K =i"d,.v—i"ti..o, ! 



.hx- 



eyeo'ff— =i"do—i"d^a.', eyd.o-^=i"6o-i"d,x, 



<* A^,-! S ' 2 3 ^^,. 



hx' 



1 - "^ 



, hx" 



oo-e.,o-'^—,=i"d.x—i"e.fl, Bo-ey-^=i"d,o—i"d,x. y 



- hx- ■' ■' - fx- ' - ^ 



So-ey^— =l"d o-i"6,x, 6o%o--tf- =i"d..o-i"d,T. 



We now eome to a group of an entirely different kind, 



I" {Ox e,x) + (I'fxf = r(e,o e^o), 

 I '{dx o.,x) + (Tffxf = I' ' (00 ey) , 

 T'iQx e^)+{i'hx) = I" (So ey. 



These formula) M'erc discovered by Meyer, who gave them, in the thirty- 

 seventh volume of Crelle's Journal, under the following form :— 



Ox 



+ 



e{x+l)~~ exe(x+lj 







_2 



Q'x Wx 



H" 





e"^ 



e^ 



Qx 



B-tx+l) '^ exB.fx+l\' 



If , 9"0 



where the reader may sec them fully demonstrated from the ' Fundamenta 

 Nova ' of Jacobi. 



They are applied by him in a memoir in the fifty-sixth volume of Crelle, 

 entitled "Ueber die rati onaleVerbindungen der elliptischen Transcendenten," to 



prove that if 'U=—f r. _^l_ir -where a is supposed a function of q, then 



^ (l+«sm (/))'« ■^■^ ^' 



Zx . U can be determined from U by simple integration with respect to (x) 



onlv. 



